# Submatrix localization via message passing

Bruce Hajek, Yihong Wu, Jiaming Xu.

Year: 2018, Volume: 18, Issue: 186, Pages: 1−52

#### Abstract

The principal submatrix localization problem deals with recovering a $K\times K$ principal submatrix of elevated mean $\mu$ in a large $n\times n$ symmetric matrix subject to additive standard Gaussian noise, or more generally, mean zero, variance one, subgaussian noise. This problem serves as a prototypical example for community detection, in which the community corresponds to the support of the submatrix. The main result of this paper is that in the regime $\Omega(\sqrt{n}) \leq K \leq o(n)$, the support of the submatrix can be weakly recovered (with $o(K)$ misclassification errors on average) by an optimized message passing algorithm if $\lambda = \mu^2K^2/n$, the signal-to-noise ratio, exceeds $1/e$. This extends a result by Deshpande and Montanari previously obtained for $K=\Theta(\sqrt{n})$ and $\mu=\Theta(1).$ In addition, the algorithm can be combined with a voting procedure to achieve the information-theoretic limit of exact recovery with sharp constants for all $K \geq \frac{n}{\log n} (\frac{1}{8e} + o(1))$. The total running time of the algorithm is $O(n^2\log n)$. Another version of the submatrix localization problem, known as noisy biclustering, aims to recover a $K_1\times K_2$ submatrix of elevated mean $\mu$ in a large $n_1\times n_2$ Gaussian matrix. The optimized message passing algorithm and its analysis are adapted to the bicluster problem assuming $\Omega(\sqrt{n_i}) \leq K_i \leq o(n_i)$ and $K_1\asymp K_2.$ A sharp information-theoretic condition for the weak recovery of both clusters is also identified.